We give lower bounds for the class number in algebraic function fields defined over finite fields, which improve the Lachaud - Martin-Deschamps bounds. We give examples of towers of algebraic function fields having a large class number and we present related problems.

A new family of towers over any non-prime finite field is presented. These towers are recursively defined, and they have a surprisingly large limit. For quadratic base fields, the Drinfeld-Vladut bound is attained; for cubic base fields, the Zink bound is attained. For all other finite base fields, the limit of the tower is much larger than the limits of previously known families of function fields. Hence we obtain a significant improvement of lower bounds for Ihara's quantity A(q), for all q=p^n with odd n > 3.

Cascudo: Arithmetic codices

In this talk I will present the notion of arithmetic codices, recently introduced in joint work with Ronald Cramer (CWI Amsterdam) and Chaoping Xing (NTU Singapore). Arithmetic codices have important applications in cryptography (in secure multiparty and two party computation) and in the complexity of multiplications in extension fields. Good towers of function fields provide the only known way of constructing 'asymptotically good families' of codices. In addition, I will motivate our introduction and study of what we have called the 'torsion limit' of a tower.

Mak: On lower bounds for the Ihara constants A(2) and A(3)

Let F_q be the finite field with q elements. The Ihara constant A(q) is an asymptotic measure of how many points a curve over F_q can have compared to its genus when the genus goes to infinity. In this talk, we will describe how to improve the best known lower bounds for A(2) and A(3). This is joint work with I. Duursma.

Perret: Some remarks on the loci of decomposed and of singular points in recursive towers

With the help of a graph closely related to Beelen's one and to intersection theory on surfaces, we will give some insights into the sets of decomposed points and of singular points in good recursive towers.

This workshop is supported by Sabanci University. The visit of A. Garcia to Sabanci University is supported by Tübitak.